1. Field of the Invention
The present invention relates to systems and methods for simulating a single-phase fluid flow enclosed by a deformable boundary while enforcing curvature related boundary conditions.
2. Description of the Related Art
The present invention is directed towards the study of single-phase fluids. A single-phase fluid, as used in the present invention refers to a pure fluid or a mixture. A boundary encapsulates this single-phase fluid, over time the boundary may evolve and move through space. The solution domain is limited to an area delimited by the boundary. For the purposes of the present invention, the effect of an interface, if any that does form between elements of such a mixture may be ignored when considering the motion of the boundary.
Motion of the boundary may be governed by one or more governing equations. The boundary may be defined by a set of markers. For some systems the governing equations may include terms related to the curvature of the boundary. Prior art methods have calculated the curvature of the boundary from a cubic spline fitted to the markers that define the boundary. Small errors in the positions of the marker can cause errors in the second derivatives, which can create variations in the curvature. Under certain conditions, this variation may increase in magnitude as the boundary evolves in time. This can lead to numerically unstable simulations, which exhibit unrealistic oscillations. The present invention is directed towards minimizing this effect and creating numerically stable simulations.
Prior art methods have attempted to address the problems that arise from defining the boundary using markers by employing multi-phase fluid simulation methods (e.g., front tracking methods). Front tracking methods have not been developed for single-phase fluid flow because there is no fluid on the other side of the boundary thus creating a jump condition in the system variable and their derivatives.
Problems arise when handling the boundary between the two fluids. Each fluid is characterized by a set of system variables (i.e., density, viscosity, and pressure). A jump condition necessarily exists at the boundary. This jump condition is represented by a step wise change in the system variables. This jump condition results in a discontinuity in system variables and their derivatives at the boundary. This discontinuity can prevent numerical solutions from converging.
Prior art methods have attempted to address this issue by smoothing system variables that cross the boundary. Smoothing these variables brings its own problems. Another problem with prior art methods is inability to conserve mass. The present invention is an attempt to address these problems with the prior art.